On groups with square-free gcd of character degree and codegree
Abstract
Let G be a finite group and χ be an irreducible character of G. The codegree of χ is defined as χc(1)=|G:kerχ|χ(1). In a paper by Gao, Wang, and Chen [Citation3], it was shown that there is no finite group G satisfying the condition that gcd(χ(1),χc(1)) is prime for all χ∈Irr(G)#. We generalize this theorem by solving one of Guohua Qian’s unsolved problems on character codegrees. In Qian’s survey article [Citation9], he inquires about the structure of non-solvable finite groups with square-free gcd instead. In particular, we prove that if G is such that gcd(χ(1),χc(1)) is square-free for every irreducible character χ, then G/Sol(G) is isomorphic to one among a particular list of almost simple groups.
Fields of Study & Subjects
Finite Group Theory Research
Advanced Algebra and Geometry
Graph theory and applications
Publication details
- Journal / source
- Communications in Algebra
- Volume
- 54
- Issue
- 12
- Pages
- 3692–3698
- Publication date
- 13 January 2026
- Language
- EN